Composition Operators On De Branges-rovnyak Spaces Associated To A Rational (Not Inner) Function

In this paper, we characterize the boundedness, the compactness and the Hilbert-Schmidt property for composition operators acting from a de Branges-Rovnyak space $\mathcal{H}(b)$ into itself, when $b$ is a rational function in the closed unit ball of $H^{\mathrm{\infty }}$ (but not a finite Blaschke product). In particular, we extend some of the results obtained by D. Sarason and J.N. Silva in the context of local Dirichlet spaces.